What is the derivative of log_a(x)?

The derivative of ( \log_a(x) ), which represents the logarithm of ( x ) to the base ( a ), can be derived using the change of base formula and the properties of logarithmic differentiation.

To find the derivative, we can start by rewriting the logarithm in terms of the natural logarithm (base ( e )). The change of base formula states that:

[

\log_a(x) = \frac{\ln(x)}{\ln(a)}

]

Now, to differentiate ( \log_a(x) ), we will apply the derivative rules. The derivative of ( \ln(x) ) is ( \frac{1}{x} ), so applying the quotient rule yields:

[

\frac{d}{dx}\left(\log_a(x)\right) = \frac{\frac{d}{dx}(\ln(x)) \cdot \ln(a) - \ln(x) \cdot \frac{d}{dx}(\ln(a))}{(\ln(a))^2}

]

Since ( \ln(a) ) is a constant, its derivative is ( 0 ). Thus, the expression simplifies to:

[

\frac{1 \cdot \